Finite Group Databasebeta

← #13156034

Subgroups — #13156034

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic, complemented
3C2×C3×#3715C211yesmaximal, characteristic, complemented
4C2×C3×#3713C211yesmaximal, characteristic
5#11722078C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10591567—31nomaximal
9#10591476—31nomaximal
10#10591431—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×D3×C3⋊Dic3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3715—22no—
23C3×#3713—22no—
24D4×C3⋊Dic3—31no—
25C3×#947—31no—
26C2×C3×#215—31no—
27C3×#936—31no—
28C2×C3×#214—31no—
29C2×C3×#213—31no—
30C2×C3×#213—31no—
31Dic3×D12—31no—
32Dic3×D12—31no—
33Dic3×D12—61no—
34#63863D311yescharacteristic, complemented
35C2×#3715D311yescharacteristic, complemented
36C2×#3713D311yescharacteristic
37#63662—31no—
38#63659—31no—
39#63572—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63563—31no—
46#63563—31no—
47#63563—61no—
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D412yes—
59C32×Dic6—21no—
60C32×Dic6—21no—
61D3×C3⋊Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5165—91no—
65#5153—91no—
66#5057—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C22×C3⋊Dic3—61no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—62no—
83#3757—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—61no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—121no—
91C4×C3⋊Dic3C2×D311yescharacteristic
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#217—61no—
95C3×#217—61no—
96C3×#215—62no—
97C3×#214—62no—
98C3×#213—62no—
99C3×#213—62no—
100C3×(C2×D3)⋊C4—61no—
101C3×(C2×D3)⋊C4—61no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—61no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8—61no—
111(C2×D3)⋊Dic3—21no—
112(C2×D3)⋊Dic3—21no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—62no—
116#3715—22no—
117#3713—22no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—22no—
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3⋊(C3⋊Dic3)—41no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—21no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133C3×#124—91no—
134#947—31no—
135C2×#215—31no—
136#936—31no—
137D4×Dic3—91no—
138D4×Dic3—91no—
139D4×Dic3—181no—
140C2×#214—31no—
141C2×#213—31no—
142C2×#213—31no—
143C4×D12—91no—
144#835—91no—
145#835—91no—
146#835—181no—
147#832—91no—
148#832—91no—
149#832—181no—
150#821—91no—
151#816—91no—
152Dic3⋊C8—91no—
153Dic3⋊C8—91no—
154C4⋊(C3⋊C8)—91no—
155C23×C32—61no—
156C23×C32—61no—
157C22×C3×D3—22no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—41no—
163C32×Q8—61no—
164C32×Q8—61no—
165C32×D4—32no—
166C32×D4—61no—
167C32×D4—61no—
168C32×D4—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—61no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—62no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—121no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—122no—
184C2×C3⋊Dic3—122no—
185C2×C3⋊Dic3#6711yescharacteristic
186C2×C3⋊Dic3#6711yescharacteristic
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C3×D12—22no—
197C3×D12—22no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—22no—
205C3×D12—41no—
206C3×D12#6712yes—
207C3×D12#6712yes—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214D3×Dic3—124no—
215D3×Dic3—122no—
216D3×Dic3—122no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—121no—
224C3×C3⋊C8—121no—
225#463—271noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33AGL(1,ℤ/8)11yescharacteristic
229C32×D3—42no—
230C32×D3—81no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—61no—
239C2×C3×D4—61no—
240C22×Dic3—181no—
241C22×Dic3—181no—
242C22×Dic3—182no—
243C2×D12D3211yescharacteristic
244C2×D12D3211yescharacteristic
245C2×C4×D3—181no—
246C2×Dic6D3211yescharacteristic
247C3×SD16—182no—
248C3×D8—182no—
249C2×C3×C8—91no—
250C2×C3×C8—91no—
251C3×C22⋊C4—181no—
252C3×C22⋊C4—181no—
253C3×C42—91no—
254#217—61no—
255#217—61no—
256#217—181no—
257#217—181no—
258#217—182no—
259#215—62no—
260#214—62no—
261#213—62no—
262#213—62no—
263(C2×D3)⋊C4—61no—
264(C2×D3)⋊C4—61no—
265(C2×D3)⋊C4—181no—
266C4⋊Dic3—91no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—181no—
270C4⋊Dic3—181no—
271C4⋊Dic3—181no—
272C4⋊Dic3—361no—
273C4×Dic3—31no—
274C4×Dic3—31no—
275C4×Dic3—61no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8—61no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—62no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32—121no—
292C22×C32D3×D411yescharacteristic
293C22×C32D3×D411yescharacteristic
294C22×C32C2×#6711yescharacteristic
295C2×C3×D3—24no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—42no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—81no—
312C2×C3×D3—81no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—121no—
322C32×C4D3×D411yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23711yescharacteristic
325C32×C4#23711yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C3⋊Dic3—21no—
329C3⋊Dic3—21no—
330C3⋊Dic3—121no—
331C3⋊Dic3—122no—
332C3⋊Dic3—121no—
333C3⋊Dic3—121no—
334C3⋊Dic3—241no—
335C3⋊Dic3—241no—
336C3⋊Dic3—242no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C3×Dic3—121no—
346C2×SD16—91no—
347C2×D8—91no—
348#124—91no—
349C4×D4—271no—
350C4⋊C8—271no—
351#103—271no—
352#102—271no—
353C33#46311yescharacteristic, complemented, Sylow
354C23×C3—62no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—61no—
358C23×C3—61no—
359C23×C3—121no—
360C22×D3—21no—
361C22×D3—21no—
362C3×Q8—62no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×Q8—121no—
368C3×D4—32no—
369C3×D4—32no—
370C3×D4—62no—
371C3×D4—62no—
372C3×D4—62no—
373C3×D4—61no—
374C3×D4—61no—
375C3×D4—61no—
376C3×D4—61no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—121no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—121no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—182no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402C2×Dic3—181no—
403C2×Dic3—361no—
404C2×Dic3—361no—
405C2×Dic3—362no—
406C2×Dic3—364no—
407C2×Dic3—362no—
408C2×Dic3—362no—
409D12—21no—
410D12—21no—
411D12—21no—
412D12#62512yes—
413C4×D3—362no—
414Dic6—21no—
415Dic6—21no—
416C3×C8—181no—
417C3×C8—181no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—121no—
423C3⋊C8—121no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—122no—
437C2×C32—241no—
438C2×C32#91111yescharacteristic
439C2×C32#91211yescharacteristic
440C2×C32#91411yescharacteristic
441C2×C32#91611yescharacteristic
442C2×C32#93811yescharacteristic
443C2×C32#93911yescharacteristic
444C2×C32#93911yescharacteristic
445C2×C32#94911yescharacteristic
446C2×C32#95011yescharacteristic
447C3×D3—44no—
448C3×D3—42no—
449C3×D3—42no—
450C3×D3—82no—
451C3×D3—81no—
452C3×D3—81no—
453C2×Q8—31no—
454C2×D4—31no—
455C2×D4—31no—
456C22×C4—541no—
457SD16—182no—
458D8—182no—
459C2×C8—91no—
460C2×C8—91no—
461C4⋊C4—271no—
462C22⋊C4—181no—
463C22⋊C4—181no—
464C22⋊C4—541no—
465C42—91no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—121no—
485C22×C3—121no—
486C22×C3—241no—
487C22×C3—241no—
488C22×C3D3×#6711yescharacteristic
489C22×C3D3×#6711yescharacteristic
490C22×C3#381111yescharacteristic
491C2×D3—22no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—122no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—181no—
509C3×C4—181no—
510C3×C4#380511yescharacteristic
511C3×C4#380511yescharacteristic
512C3×C4#380511yescharacteristic
513C3×C4C2×#62511yescharacteristic
514C3×C4D3×#6711yescharacteristic
515C3×C4D3×#6711yescharacteristic
516Dic3—41no—
517Dic3—62no—
518Dic3—61no—
519Dic3—61no—
520Dic3—61no—
521Dic3—61no—
522Dic3—121no—
523Dic3—362no—
524Dic3—361no—
525Dic3—361no—
526Dic3—361no—
527Dic3—721no—
528Dic3—721no—
529Dic3—722no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#505711yescharacteristic, complemented
535C32#515311yescharacteristic, complemented
536C32#516511yescharacteristic, complemented
537C23—61no—
538C23—61no—
539Q8—61no—
540Q8—61no—
541D4—32no—
542D4—61no—
543D4—61no—
544D4—61no—
545C2×C4—61no—
546C2×C4—91no—
547C2×C4—91no—
548C2×C4—541no—
549C2×C4—1082no—
550C2×C4D3311yescharacteristic
551C8—181no—
552C8—181no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—124no—
566C2×C3—122no—
567C2×C3—122no—
568C2×C3—242no—
569C2×C3—241no—
570C2×C3—241no—
571C2×C3#6392511yescharacteristic
572C2×C3#6393311yescharacteristic
573C2×C3#6393611yescharacteristic
574C2×C3#6393711yescharacteristic
575C2×C3#6393911yescharacteristic
576C2×C3#6394711yescharacteristic
577C2×C3#6397011yescharacteristic
578C2×C3#6397011yescharacteristic
579C2×C3#6398011yescharacteristic
580D3—42no—
581D3—81no—
582C22—62no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22#8895211yescharacteristic
589C4—121no—
590C4—181no—
591C4—181no—
592C4—1081no—
593C4D3×#62511yescharacteristic
594C4#8896311yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059143111yescharacteristic, complemented
600C3#1059147611yescharacteristic, complemented
601C3#1059156711yescharacteristic, complemented
602C2—122no—
603C2—241no—
604C2#1172285711yescharacteristic
605C2#1172287411yescharacteristic
606C2#1172295311yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. D12×C3⋊Dic3C2semidirect
2. C3×#63863C2semidirect
3. C2×C3×#3715C2semidirect
5. #11722078C2semidirect
6. #11722038C2semidirect
7. #11722032C2semidirect
34. #63863D3semidirect
35. C2×#3715D3semidirect
353. C33#463semidirect
534. C32#5057semidirect
535. C32#5153semidirect
536. C32#5165semidirect
599. C3#10591431semidirect
600. C3#10591476semidirect
601. C3#10591567semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D12×C3⋊Dic32
C3×#638632
C2×C3×#37152
C2×C3×#37132
#117220782
#117220382
#117220322
#105915673
#105914763
#105914313